A charge Q = -610 NC (-6.1 × 10^-7 C) is uniformly distributed on a ring of 2.4-m radius. A point charge q = +480 nC (+4.8 × 10^-7 C) is fixed at the center of the ring, as shown in the figure. An electron is projected from infinity toward the ring along the axis of the ring. This electron comes to a momentary halt at a point on the axis that is 5.0 m from the center of the ring. What is the initial speed of the electron at infinity? (e = 1.60 × 10^-19 C, k = 1/4πε_0 = 8.99 × 10^9 N · m^2/C^2, m_e = 9.11 x 10-31 kg)

Question

A charge Q = -610 NC (-6.1 × 10^-7 C) is uniformly distributed on a ring of 2.4-m radius. A point charge q = +480 nC (+4.8 × 10^-7 C) is fixed at the center of the ring, as shown in the figure. An electron is projected from infinity toward the ring along the axis of the ring. This electron comes to a momentary halt at a point on the axis that is 5.0 m from the center of the ring. What is the initial speed of the electron at infinity? (e = 1.60 × 10^-19 C, k = 1/4πε_0 = 8.99 × 10^9 N · m^2/C^2, m_e = 9.11 x 10-31 kg)

a. 2.2 × 10^6 m/s
b. 3.4 × 10^6 m/s
c. 6.6 × 10^6 m/s
d. 1.1 × 10^6 m/s
e. 4.5 × 10^6 m/s

Apparently, the correct answer to this question is 6.6 × 10^6 m/s. I saw a solution on how that answer is achieved, but despite following all the steps exactly, I ended up with a completely different answer. I don't know what I am doing wrong and it is frustrating me.

Answer 1

To determine the initial speed of the electron at infinity, we need to calculate the potential energy of the electron as it approaches the ring and equate it to the kinetic energy of the electron at infinity.

Here is a step-by-step approach to solve the problem:

Step 1: Calculate the potential energy of the electron due to the fixed charge q at the center of the ring.
The potential energy due to the point charge is given by:
U = k * (q1 * q2) / r
where k is the electrostatic constant (k = 8.99 × 10^9 N · m^2/C^2), q1 is the charge of the electron, q2 is the charge at the center of the ring, and r is the distance between the charges.

U = (8.99 × 10^9 N · m^2/C^2) * (1.60 × 10^-19 C) * (4.8 × 10^-7 C) / 5.0 m

Step 2: Determine the initial potential energy of the electron at infinity. At infinity, the potential energy is zero since the charges are effectively infinitely far apart.

Ui = 0

Step 3: Use the conservation of energy to equate the potential energy at infinity with the potential energy at the point of closest approach on the ring.

Ui = U

Step 4: Rewrite the equation in terms of the initial kinetic energy of the electron at infinity (K)

K + Ui = U

Step 5: Solve for K, the initial kinetic energy of the electron at infinity.

K = U - Ui

Step 6: Calculate the initial speed of the electron using the equation for kinetic energy:

K = (1/2) * m_e * v^2
where m_e is the mass of the electron (9.11 x 10^-31 kg) and v is the initial speed of the electron.

Step 7: Rearrange the equation to solve for v:

v = sqrt((2 * K) / m_e)

Step 8: Substitute the calculated values of U, Ui, and m_e into the equation and find the initial speed of the electron at infinity.

Make sure to verify that you have entered the values correctly and performed all the calculations accurately. If your calculated answer is different from the expected answer, double-check your calculations and units to identify any mistakes that may have been made.